# Numerical Experiments

## 1. Problem Statement

We have a geometry splitted in 9 subparts :

Geometry

image::example-thermal-block-heat-transfer.png

Given $\mu \in (\mu_1,...\mu_P) \in \mathcal{D}^\mu\equiv [\mu^{\text{min}},\mu^{\text{max}}]^P$, evaluate (recall that $\ell = f$)
$s(\mu) = f(u(\mu))]$ where $u(\mu) \in X \equiv \{ v \in H^1(\Omega), v|_{\Gamma_{\text{top}}} = 0\}$ satisfies $a(u(\mu), v; \mu) = f(v; \mu) \quad \forall v \in X$ we have $P = 8$ and given $1 < \mu_r < \infty$ we set $\mu^{\mathrm{min}} = \dfrac{1}{\sqrt{\mu_r}},\quad \mu^{\mathrm{max}} = \sqrt{\mu_r}$ such that $\dfrac{\mu^{\mathrm{max}}}{\mu^{\mathrm{min}}}=\mu_r$.

Recall we are in the compliant case $\ell = f$, we have $f(v) = \displaystyle\int_{\Gamma_{0}} v\quad \forall v \in X$ and $a(u,v;\mu) = \displaystyle\sum_{i=1}^{P} \mu_i \int_{\Omega_i} \nabla u \cdot \nabla v + 1 \int_{\Omega_{P+1}} \nabla u \cdot \nabla v \quad\forall u,\ v\ \in X$ where $\Omega = \displaystyle\cup_{i=1}^{P+1} \Omega_i$.

The inner product is defined as follows $(u,v)_X = \displaystyle\sum_{i=1}^P \bar{\mu}_i \int_{\Omega_i}\nabla u \cdot \nabla v + 1 \int_{\Omega_{P+1}} \nabla u \cdot \nabla v$ where $\bar{\mu}_i$ is a reference parameter. We have readily that $a$ is

• symmetric,

• parametrically coercive $0 < \dfrac{1}{\sqrt{\mu_r}} \leq \mathrm{min}(\mu_1/\bar{\mu}_1, \ldots, \mu_P/\bar{\mu}_P,1) \leq \alpha(\mu)$,

• and continuous $\gamma(\mu) \leq \mathrm{max}(\mu_1/\bar{\mu}_1, \ldots, \mu_P/\bar{\mu}_P,1) \leq \sqrt{\mu_r} < \infty$

and the linear form $f$ is bounded.

## 2. Affine decomposition

We obtain the affine decomposition $a(u,v;\mu) = \displaystyle\sum_{q=1}^{P+1} \Theta^q(\mu) a^q(u,v)$ with \begin{aligned} \Theta^1(\mu) = \mu_1 & & a^1(u,v) = \int_{\Omega_1} \nabla u \cdot \nabla v\\ & \vdots & \\ \Theta^P(\mu) = \mu_P & & a^P(u,v) = \int_{\Omega_P} \nabla u \cdot \nabla v\\ \Theta^{P+1}(\mu) = 1 & & a^{P+1}(u,v) = \int_{\Omega_{P+1}} \nabla u \cdot \nabla v \end{aligned}

## 3. Results

• Homogeneous parameters

Maximum parameter values

image::33-max.png

Minimum parameter values

image::33-min.png

• Heteregeneous parameters

Random values

image::33-random.png

## 4. Thermal Block $P=1$

Random values

image::thermalblock-P1.png

We assume $1/\mu^{\min}_1=\mu^{\max}_1=\sqrt{\mu_r}=10$ and choose $\mathcal{N}=256$

we set $\bar{\mu}=1$ and have $\Theta^1_a(\mu)=\mu_1,\, \Theta^2_a(\mu) = 1$. Thus, $\Theta_a^{\min,\bar{\mu}}(\mu_1)=\min(\mu_1,1)\\ \Theta_a^{\max,\bar{\mu}}(\mu_1)=\max(\mu_1,1)$

hence $\theta^{\bar{\mu}}(\mu_1) = \max(\frac{1}{\mu_1},\mu_1)$ and $\theta^{\bar{\mu}}(\mu_1) \leq\sqrt{\mu_r} \forall \mu_1 \in \mathcal{D}$.

Values in [RHP2008]

Table 1. Convergence results for $P=1$
$N$ $\Delta^s_{N,\mathrm{max}}(\mu)$ $\eta^s_{N,\mathrm{ave}}$ $\eta^s_{N,\mathrm{max}}$

1

7.2084E+00

2.3417

3.3305

2

4.5371E–01

2.4858

3.6850

3

6.9652E–04

6.2195

9.8551

4

1.3744E–07

3.3219

7.2632

5

3.1140E–11

6.0789

7.0453

Note that: $\eta^s_{N,\mathrm{max}}(\mu_1) \leq \eta^s_{\mathrm{max,UB}} \equiv \sqrt{\mu_r} = 10$

• Maximum output error bound: $\Delta^s_{N,\mathrm{max}} = \max_{\mu \in \Xi_{\mathrm{train}}} \Delta^s_N(\mu)$

• Average output effectivity: $\eta^s_{N,\mathrm{ave}} = \frac{1}{\Xi_\mathrm{train}}\sum_{\mu \in \Xi_{\mathrm{train}}} \eta^s_N(\mu)$

• Maximum output effectivity: $\eta^s_{N,\mathrm{max}} = \max_{\mu \in \Xi_{\mathrm{train}}} \eta^s_N(\mu)$

## 5. Example Thermal Block $P=8$

• Configuration :

• 47 600 dofs;

• Preconditionner : LU – Solver : MUMPS

• $\Xi :$ parameter sampling of dimension 1 000.

• Plot $\max_{\mu \in \Xi} \frac{ |s^{\mathcal{N}}(\mu)-s_N(\mu)|}{s^{\mathcal{N}}(\mu)}$

image::Relative-error.png

Thermal Block $P=8$

• More parameters there are, more rich the problem is;

• Notations :

• $e^i(\mu)$ is the relative error on the output when $i$ parameters vary.

image::Relative-error-2.png